The long term goals of this research program are to develop statistical methodology for the analysis and biomedical data that is based on non- and semi-parametric curve estimation theory, and to develop tools for parametric model building. Computer simulation studies will be executed to verify that the asymptotic theory developed is useful for interference with sample sizes typically encountered in biomedical data sets described in this proposal. The following problems will be studied during the proposed funding period: 1. Estimation in the Additive Model for Non-parametric Regression: An algorithm is proposed here for fitting the additive model in multiple regression. Consistency of the estimators should hold without restricting dependencies among the co-variates. 2. Non-parametric Regression Analysis of Multi-variate Longitudinal Data. a. This research project is motivated by a data set that was obtained from Phase I study to determine the safety of Droloxifene in patients with advanced metastatic breast cancer. The levels of several hormones were monitored at the time of entry to the study and repeatedly thereafter. This is a continuation of research reported in Staniswalis and Lee (1997). The smooth non-parametric estimators of the covariance function will be studied further. Development of methods for a canonical correlation analysis for random curves is proposed to characterize associations among the four different hormones over time, and to determine if there is a dependence on the dose of Droloxifene. b. Redundancy analysis is used when it is of interest to predict one set of variables with a predictor constructed from another set of variables. For this specific aim, this standard multi-variate methodology of redundancy analysis for vectors will be extended to understand how a collection of random curves could be used to best predict another collection of random curves.